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Mirrors > Home > ILE Home > Th. List > fzo0dvdseq | GIF version |
Description: Zero is the only one of the first 𝐴 nonnegative integers that is divisible by 𝐴. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
fzo0dvdseq | ⊢ (𝐵 ∈ (0..^𝐴) → (𝐴 ∥ 𝐵 ↔ 𝐵 = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzolt2 10081 | . . . . . . 7 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 < 𝐴) | |
2 | elfzoelz 10072 | . . . . . . . 8 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 ∈ ℤ) | |
3 | elfzoel2 10071 | . . . . . . . 8 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐴 ∈ ℤ) | |
4 | zltnle 9228 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵)) | |
5 | 2, 3, 4 | syl2anc 409 | . . . . . . 7 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵)) |
6 | 1, 5 | mpbid 146 | . . . . . 6 ⊢ (𝐵 ∈ (0..^𝐴) → ¬ 𝐴 ≤ 𝐵) |
7 | 6 | adantr 274 | . . . . 5 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐴 ∥ 𝐵) → ¬ 𝐴 ≤ 𝐵) |
8 | 3 | adantr 274 | . . . . . . 7 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐴 ∈ ℤ) |
9 | elfzonn0 10111 | . . . . . . . . . 10 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐵 ∈ ℕ0) | |
10 | 9 | adantr 274 | . . . . . . . . 9 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℕ0) |
11 | simpr 109 | . . . . . . . . 9 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ≠ 0) | |
12 | eldifsn 3697 | . . . . . . . . 9 ⊢ (𝐵 ∈ (ℕ0 ∖ {0}) ↔ (𝐵 ∈ ℕ0 ∧ 𝐵 ≠ 0)) | |
13 | 10, 11, 12 | sylanbrc 414 | . . . . . . . 8 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ∈ (ℕ0 ∖ {0})) |
14 | dfn2 9118 | . . . . . . . 8 ⊢ ℕ = (ℕ0 ∖ {0}) | |
15 | 13, 14 | eleqtrrdi 2258 | . . . . . . 7 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℕ) |
16 | dvdsle 11767 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 ∥ 𝐵 → 𝐴 ≤ 𝐵)) | |
17 | 8, 15, 16 | syl2anc 409 | . . . . . 6 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐵 ≠ 0) → (𝐴 ∥ 𝐵 → 𝐴 ≤ 𝐵)) |
18 | 17 | impancom 258 | . . . . 5 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐴 ∥ 𝐵) → (𝐵 ≠ 0 → 𝐴 ≤ 𝐵)) |
19 | 7, 18 | mtod 653 | . . . 4 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐴 ∥ 𝐵) → ¬ 𝐵 ≠ 0) |
20 | 0z 9193 | . . . . . . . 8 ⊢ 0 ∈ ℤ | |
21 | zdceq 9257 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝐵 = 0) | |
22 | 20, 21 | mpan2 422 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → DECID 𝐵 = 0) |
23 | nnedc 2339 | . . . . . . 7 ⊢ (DECID 𝐵 = 0 → (¬ 𝐵 ≠ 0 ↔ 𝐵 = 0)) | |
24 | 22, 23 | syl 14 | . . . . . 6 ⊢ (𝐵 ∈ ℤ → (¬ 𝐵 ≠ 0 ↔ 𝐵 = 0)) |
25 | 2, 24 | syl 14 | . . . . 5 ⊢ (𝐵 ∈ (0..^𝐴) → (¬ 𝐵 ≠ 0 ↔ 𝐵 = 0)) |
26 | 25 | adantr 274 | . . . 4 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐴 ∥ 𝐵) → (¬ 𝐵 ≠ 0 ↔ 𝐵 = 0)) |
27 | 19, 26 | mpbid 146 | . . 3 ⊢ ((𝐵 ∈ (0..^𝐴) ∧ 𝐴 ∥ 𝐵) → 𝐵 = 0) |
28 | 27 | ex 114 | . 2 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐴 ∥ 𝐵 → 𝐵 = 0)) |
29 | dvds0 11732 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∥ 0) | |
30 | 3, 29 | syl 14 | . . 3 ⊢ (𝐵 ∈ (0..^𝐴) → 𝐴 ∥ 0) |
31 | breq2 3980 | . . 3 ⊢ (𝐵 = 0 → (𝐴 ∥ 𝐵 ↔ 𝐴 ∥ 0)) | |
32 | 30, 31 | syl5ibrcom 156 | . 2 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐵 = 0 → 𝐴 ∥ 𝐵)) |
33 | 28, 32 | impbid 128 | 1 ⊢ (𝐵 ∈ (0..^𝐴) → (𝐴 ∥ 𝐵 ↔ 𝐵 = 0)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 824 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 ∖ cdif 3108 {csn 3570 class class class wbr 3976 (class class class)co 5836 0cc0 7744 < clt 7924 ≤ cle 7925 ℕcn 8848 ℕ0cn0 9105 ℤcz 9182 ..^cfzo 10067 ∥ cdvds 11713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-fz 9936 df-fzo 10068 df-dvds 11714 |
This theorem is referenced by: fzocongeq 11781 |
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